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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 | import Mathlib
import OAI.Probability.LogConcave.Numerics.VelocityWorkCoefficient
namespace OAI
section
noncomputable section
namespace LogConcaveSampling
open Filter Quadrature MeanTree
open scoped Topology
lemma logMeshCount_rate {T : ℕ → ℝ} {b : ℝ} (hb : 0≤b)
(hT0 : ∀ᶠ d : ℕ in atTop,0<T d) (hT1 : ∀ᶠ d : ℕ in atTop,T d<1)
(hL : LogPowerRate (fun d => logMeshLength (T d)) 0) :
LogPowerRate (fun d => (logMeshCount (T d) ((d:ℝ)^(-b)):ℝ)) (-b) := by
have hr := hL.mul (LogPowerRate.inv_power b)
simp only [zero_add] at hr
apply LogPowerRate.of_le (hr.add ((LogPowerRate.const 1).mono (by linarith)))
· exact Eventually.of_forall (fun _ => Nat.cast_nonneg _)
· filter_upwards [hT0,hT1,eventually_ge_atTop (1:ℕ)] with d hd0 hd1 hd
have hp : (0:ℝ)<d := by exact_mod_cast hd
simpa only [div_eq_mul_inv] using (logMeshCount_bound hd0 hd1 (Real.rpow_pos_of_pos hp (-b))).le
lemma probabilityNode_card_rate {T : ℕ → ℝ} {b : ℝ} (hb : 0≤b) (n : ℕ)
(hT0 : ∀ᶠ d : ℕ in atTop,0<T d) (hT1 : ∀ᶠ d : ℕ in atTop,T d<1)
(hL : LogPowerRate (fun d => logMeshLength (T d)) 0) :
LogPowerRate (fun d => (Fintype.card (ProbabilityNode (T d) ((d:ℝ)^(-b)) n):ℝ)) (-b) := by
simpa only [probabilityNode_card,Nat.cast_mul,Nat.cast_add,Nat.cast_one] using
(logMeshCount_rate hb hT0 hT1 hL).mul_const ((n:ℝ)+1)
theorem meanWork_rate {p : ℕ → ℕ} {b : ℝ} (hb : 0≤b)
(hp : LogPowerRate (fun d => (p d:ℝ)) (-b)) (q m N nc Nc : ℕ) :
LogPowerRate (fun d => (meanWork (p d) q m N nc Nc 0 0 0:ℝ)) (-((Nc*(2*N+2):ℕ):ℝ)*b) := by
have hr := (hp.add ((LogPowerRate.const 1).mono (by linarith))).pow (Nc*(2*N+2))
have he := hr.const_mul (((nc+1)*((nc+1)*velocityWorkCoefficient q m N)^Nc:ℕ):ℝ)
simp only [mul_neg,neg_mul] at he ⊢
apply LogPowerRate.of_le he
· exact Eventually.of_forall (fun _ => Nat.cast_nonneg _)
· apply Eventually.of_forall
intro d
exact_mod_cast meanWork_polynomial (p d) q m N nc Nc
lemma sampleWork_rate {p : ℕ → ℕ} {b : ℝ} (hb : 0≤b)
(hp : LogPowerRate (fun d => (p d:ℝ)) (-b)) (N : ℕ) :
LogPowerRate (fun d => (flowWork (p d) N 0 0:ℝ)) (-(N:ℝ)*b) := by
have he := (hp.add ((LogPowerRate.const 1).mono (by linarith))).pow N
simp only [mul_neg,neg_mul] at he ⊢
apply LogPowerRate.of_le he
· exact Eventually.of_forall (fun _ => Nat.cast_nonneg _)
· apply Eventually.of_forall
intro d
have hh := flowWork_bound (p d) N 0
simp only [zero_add,one_mul] at hh
have : flowWork (p d) N 0 0≤(p d+1)^N := by omega
exact_mod_cast this
lemma routineBudget_rate {c : ℕ → ℕ} {a : ℝ} (ha : 0≤a)
(hc : LogPowerRate (fun d => (c d:ℝ)) (-a)) (rank : ℕ) :
LogPowerRate (fun d => ((2*(40*c d+40)^rank:ℕ):ℝ)) (-(rank:ℝ)*a) := by
have he := (((hc.const_mul 40).add ((LogPowerRate.const 40).mono (by linarith))).pow rank).const_mul 2
simp only [mul_neg,neg_mul] at he ⊢
simpa only [Nat.cast_mul,Nat.cast_pow,Nat.cast_add,Nat.cast_ofNat] using he
end LogConcaveSampling
end
end
section
noncomputable section
namespace LogConcaveSampling
open Filter Quadrature
open scoped Topology
lemma sampleCorrelation_antitone {a b : ℝ} (ha : 0≤a) (hab : a≤b) :
sampleCorrelation b≤ sampleCorrelation a := by
unfold sampleCorrelation
apply (inv_le_inv₀ (Real.sqrt_pos.mpr (by positivity))
(Real.sqrt_pos.mpr (by positivity))).mpr
exact Real.sqrt_le_sqrt (by nlinarith)
lemma logMeshCount_mono {T U h : ℝ} (hT : T≤U) (hU : U<1) (hh : 0<h) :
logMeshCount T h≤logMeshCount U h := by
apply Nat.ceil_mono
apply div_le_div_of_nonneg_right _ hh.le
unfold logMeshLength
have he := Real.log_le_log (by linarith : 0<1-U) (by linarith : 1-U≤1-T)
linarith
lemma sampleCorrelation_radius_lower {η : ℝ} (hη : 0<η) (hη1 : η≤1) :
η/4≤Real.sqrt (1-(sampleCorrelation η)^2) := by
have hp := sampleCorrelation_properties hη hη1
have ht : 0<sampleCorrelation η := by linarith
have he : Real.sqrt (1-(sampleCorrelation η)^2)=sampleCorrelation η*(η/2) := by
have h := congrArg (fun z : ℝ => sampleCorrelation η*z) hp.2.2
simpa only [←mul_assoc,mul_inv_cancel₀ ht.ne',one_mul] using h
rw [he]
nlinarith
lemma logMeshLength_sample_rate {η : ℕ → ℝ} {a : ℝ}
(hη : ∀ᶠ d : ℕ in atTop,(d:ℝ)^(-a)≤η d ∧ η d≤1) :
LogPowerRate (fun d => logMeshLength (sampleCorrelation (η d))) 0 := by
have hr := ((LogPowerRate.const (Real.log 2)).add
(((LogPowerRate.log_power a).add (LogPowerRate.const (-Real.log 4))).const_mul (-2)))
apply LogPowerRate.of_le hr
· filter_upwards [hη,eventually_ge_atTop (1:ℕ)] with d hd hd1
have hp : (0:ℝ)<d := by exact_mod_cast hd1
have hpη := sampleCorrelation_properties (lt_of_lt_of_le (Real.rpow_pos_of_pos hp _) hd.1) hd.2
exact (logMeshLength_pos (by linarith [hpη.1]) hpη.2.1).le
· filter_upwards [hη,eventually_ge_atTop (1:ℕ)] with d hd hd1
have hp : (0:ℝ)<d := by exact_mod_cast hd1
have hR : 0<(d:ℝ)^(-a) := Real.rpow_pos_of_pos hp _
have hηpos := lt_of_lt_of_le hR hd.1
have hpη := sampleCorrelation_properties hηpos hd.2
have hh := sampleCorrelation_radius_lower hηpos hd.2
have hu : (d:ℝ)^(-a)/4≤Real.sqrt (1-(sampleCorrelation (η d))^2) := by linarith
have hs : 0≤1-(sampleCorrelation (η d))^2 := by nlinarith [hpη.1,hpη.2.1]
have hdR : ((d:ℝ)^(-a)/4)^2≤1-(sampleCorrelation (η d))^2 := by
have he := pow_le_pow_left₀ (by positivity : 0≤(d:ℝ)^(-a)/4) hu 2
rwa [Real.sq_sqrt hs] at he
have he := logMeshLength_le_scale (by linarith [hpη.1]) hpη.2.1
(by positivity : 0<(d:ℝ)^(-a)/4) hdR
rw [Real.log_div hR.ne' (by norm_num)] at he
linarith
end LogConcaveSampling
end
end
end OAI
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